How can I set a Shape Key value to be a driver's variable?
If you use an object in the Prop field, the path to the shape key is like this:

data.shape_keys.key_blocks["Key 1"].value

If you use a mesh, you don't have to put data there, so:

shape_keys.key_blocks["Key 1"].value



And if you use the Key Prop field with your key block there, then you can just use:

key_blocks["Key 1"].value